No you should not do this. Sometimes it is explicitly allowed on the data sheet (but not that I can see on this data sheet), and when it is, in my experience you never get as much as double the capacity.
Paralleling physically separate relays is worse again because they're not physically moving together- expect welded contacts etc. if you tried that.
If you can split the load (for example, instead of a 40A heater use two 20A heaters) then you can get an equivalent functionality.
You could think about ballasting the loads (wasting power to roughly equalize the currents) and fusing each contact separately, but I don't think that's a good idea at all.
Note that using the relay at the maximum rated current will lead to a pretty short life (only 100,000 operations for a resistive load), which might be only weeks or months if it's switching continuously. At 3HP (motor load), the life is only 1,000 operations, so at once per minute it won't last a single day.
Edit: With the added information that you're using the relay to switch effectively at a relatively low DC voltage and you're mostly concerned about carrying current.. I can't say categorically this is really a horrible idea with a single relay, but I think I'd get on the horn to the manufacturer and see if it's possible to get any buy-in. It comes down to variability in contact resistance vs. the resistance of the connections (plus whatever, hopefully balanced, resistance you add externally). When one of the contacts inevitably fails first, I think I would prefer the relay to not emit excessive amounts of smoke or flames). I think you're okay at 40A (with AgCdO contacts) given the UL508 rating, but beyond that is in question.
If you really need such a high carrying current, the Omron G7Z appears to explicitly allow paralleling the 40A contacts without derating, for 160A total capacity, but perhaps not with the blessing of safety agencies.
1) Driving 35 amps through contacts rated for 10 amps is a great way to kill the contacts.
2) The contacts will fail either by welding shut or by "burning" so that they can no longer make contact. Which is more likely depends somewhat on your load. If your load is purely resistive or has a turn-on surge (like incandescent lamps or capacitors) then welding is the more likely failure. If the load is inductive (like a motor) then burning the contacts when you try to open them is more likely.
3) In the terms you're asking - no.
4) Yes. (With this exception - if the NPN can be used to interrupt the current flow before the relay is opened, it will prevent burning the contacts. Of course, if you can do this, why are you bothering with a relay?)
5) Whoever said you can push 20A through a 10A relay is giving bad advice, and the concept that "is just the relay output will not exceed 10A" makes no sense at all.
You need a beefier relay.
Best Answer
You seem to be mixing up the ratings on the coils and on the switching contacts. Notice the ratings you quoted come from two different tables in the datasheet:
If your power supply is being used to drive the coils, then you need to consider the required coil current to switch the relay. As your item #1 says, a 10 A supply could potentially supply 100 100-mA coils simultaneously.
If your power supply is being used to drive the load (switched by the relay) then you need to worry about what current your load draws when powered by 12 V. If it does indeed draw 3A, then you could only power 3 such loads with a 10 A supply.
If one relay is being used to switch current to the coils of a bunch of other relays, then you get the limit of your item #2. The 3 A allowed through the contact of the first relay is enough to power the coils of 30 100-mA coils that are its load.
Also, if you want your system to be reliable for a long time, you'd probably want to de-rate all of these specs rather than operate the power supply or the relays at their maximum current limits. (The coils will operate at roughly the spec'ed currents due to the coil resistance and Ohm's Law)